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## Portfolio Immunization and Cash Flow Matching by Frank J. Fabozzi

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**1. **Portfolio Immunization and Cash Flow Matchingby Frank J. Fabozzi

**3. **Key Learning Outcomes Identify the risks associated with immunizing a portfolio.
Explain the ways classical immunization has been extended.
Identify the factors to consider in creating an immunized portfolio.
Describe what a contingent immunization strategy is.
Identify the key considerations in implementing a contingent immunization strategy.

**4. **Key Learning Outcomes Describe for a contingent immunization strategy the following concepts: safety net level return, excess achievable return, return achievable with an immunization strategy, and trigger point.
Discuss the three conditions for creating an immunized portfolio to satisfy multiple liabilities.
Explain the different approaches suggested for valuing defined benefit plan liabilities.

**5. **Key Learning Outcomes Illustrate how a cash flow matching strategy can be used to construct a portfolio to satisfy multiple liabilities.
Explain the advantages and disadvantages of a cash flow matching strategy relative to multiple liability immunization.
Explain a combination matching strategy.

**6. **Introduction Discuss strategies for managing bond portfolios to satisfy predetermined liabilities
Immunization
“hybrid strategy having elements of both active and passive strategies . . . Used to minimize reinvestment risk over a specified investment horizon”
Cash Flow Matching
“strategy used to construct a portfolio that will fund a schedule of liabilities from a portfolio's cash flows, with the portfolio's value diminishing to zero after payment of the last liability”

**7. **Immunization Strategy for a Single Liability Classical Immunization
“The process by which a bond portfolio is created to have an assured return for a specific time horizon irrespective of interest rate changes”
Structure a portfolio that balances the change in the value of the portfolio at the end of the investment horizon with the return from the reinvestment of portfolio cash flows (coupon and principal payments)
Loss or gain in reinvestment income = loss or gain in portfolio value

**8. **Exhibit 1Total Accumulated Value and Total Return After Fives Years for a 12-year 7.5% Bond Selling to Yield 7.5%Targeted accumulated value: $13,934,413

**9. **Immunization Strategy for a Single Liability Example
Market yield rises instantly by 200 bp from 7.5% to 9.5%
Reinvestment income will be $204,039 greater
Market value of the bond will decrease by $969,958
Net effect on accumulated value will be $765,919 less than the target accumulated value

**10. **Exhibit 2 Total Accumulated Value and Total Return After Fives Years for a 6-year 6.75% Bond Selling to Yield 7.5%Targeted accumulated value: $13,934,413

**11. **Immunization Strategy for a Single Liability The accumulated value and the total return are never less than the target accumulated value and target yield
Target accumulated value is assured regardless of what happens to market yield

**12. **Immunization Strategy for a Single Liability Duration of liability is 4.82
Duration of 6-year 6.75% bond is 4.82
To immunize a portfolio's target accumulated value against a change in the market yield, the insurance company must invest in a bond:
1) Equal duration to the liability’s duration
2) Initial cash flow from the bond equals the present value of the future liability

**13. **Immunization Strategy for a Single Liability Bonds with embedded options:
1) Portfolio’s effective duration is equal to the liability’s effective duration
2) The initial present value of the projected cash flows from the bond portfolio equal the present value of the future liability

**14. **Rebalancing an Immunized Portfolio Market yield will fluctuate over the investment horizon
Duration of the portfolio will change as the market yield changes, also, duration will change simply because of the passage of time.
A portfolio can be immunized if it is rebalanced periodically so that its duration is readjusted to the duration of the liability

**15. **Rebalancing an Immunized Portfolio Example
Investment horizon is initially 5 years and the yield is 7.5% the initial duration is equal to 4.82
After six months the investment horizon will be 4.5 years and the liability’s duration will be 4.34
Thus, the portfolio’s duration must be rebalanced so that the duration is equal to 4.34 (done every six months)

**16. **Rebalancing an Immunized Portfolio How often to rebalance?
Frequent rebalancing increases transaction costs, thus reducing the likelihood of achieving target returns
Infrequent rebalancing will result in the portfolio’s duration wandering from the target duration, thus reducing the likelihood of achieving target returns
Manager must deal with the trade-off and find a happy medium

**17. **Application Considerations In forming the immunized portfolio selection of the universe is very important
Immunization theory assumes no defaults and that the securities will only be responsive to overall changes in interest rates

**18. **Application Considerations Liquidity – the portfolio must be rebalanced over time and principal liquidated on the target investment horizon to pay the liability
Securities to be avoided:
Low credit quality
Embedded options

**19. **Extensions of Classical Immunization Theory Assumption 1
Any changes in the yield curve are parallel changes
Assumption 2
The portfolio is valued at a fixed horizon date and there is no cash inflows or outflows during the time horizon except for coupon income and reinvestment income

**20. **Extensions of Classical Immunization Theory Assumption 3
The target value of the investment is defined as the portfolio value at the horizon date if the interest rate structure does not change (i.e., no change in forward rates)

**21. **Immunization Risk A strategy was developed to handle an arbitrary interest rate change with an immunization risk measure
Can be minimized subject to the constraint that the portfolio duration be equal to the investment horizon resulting in a portfolio with minimum exposure to any interest rate movements

**22. **Immunization Risk Barbell vs. Bullet Portfolios
Bullet portfolios have bonds that mature close to the investment horizon (relatively less immunization risk)
With non-parallel shifts in interest rates (i.e. short rates decline and long rates go up) both portfolios would realize a loss but the barbell portfolio would be substantially higher
A) The lower reinvestment rates are experienced for bonds in the barbell portfolio for the shorter maturities than on the bullet portfolio (higher reinvestment risk)
B) The longer maturity bond in the barbell portfolio at the end of the investment horizon is much longer than the longer maturity bond in the bullet portfolio (same rate increase generates a greater capital loss)
IMMUNIZATION RISK = REINVESTMENT RISK

**23. **Contingent Immunization Identifying both the available immunization target rate and a lower safety net level return which with a client would be minimally satisfied (i.e. minimum required rate of return)
Key Considerations:
1) Establish well defined immunized initial and outgoing available target returns
2) Identify a suitable and immediate safety net
3) Implement an effective monitoring procedure to ensure that the safety net return is not violated

**24. **Contingent Immunization Suppose a plan sponsor is willing to accept 6% return over a 5 year investment horizon at a time when the possible immunized return is 7.5%
Cushion spread – difference between the possible return and safety net return
The greater the cushion spread the more the manager room the manager has for an active investment policy

**25. **Contingent Immunization Required terminal value = I(1+s/2)2H
I = initial portfolio value
s = safety net rate
H = number of years in the investment horizon
Trigger point – the yield level at which the immunization mode becomes necessary

**26. **Controlling and Monitoring the Strategy It is important to measure the return achievable with an immunization strategy because this is the yield measure that would be realized, if at any point and time, the current value of the portfolio is immunized at the prevailing market rate

**27. **Immunizing for Multiple Liabilities More often than having a single liability payable at a horizon date, investment funds have to pay a number of liabilities a no single horizon corresponds to the schedule of liabilities
Just having the portfolio duration match the liabilities duration is not sufficient for immunizing multiple liabilities
Two strategies can be employed in this case:
An extension of the single period immunization strategy
Cash flow matching strategy

**28. **Conditions for Immunizing Multiple Liabilities Necessary conditions to assure multiple liability immunization in the case of parallel rate shifts are as follows:
1) Present value of the assets must equal the present value of the liabilities
2) The composite portfolio duration must equal the composite liabilities duration
3) The distribution of durations of individual assets in the portfolio must have a wider range than the distribution of the liabilities

**29. **Immunizing Defined Benefit Obligations Sponsors of defined benefit plans represent the major potential users of the multiple liability immunization strategy
Liabilities should be discounted using spot rates that match the horizon of each liability
Two Yield Curves proposed for valuing liabilities based on market yields and multiple interest rates:
Treasury spot rate curve
Treasury yield curve plus a spread

**30. **Cash Flow Matching for Multiple Liabilities Cash Flow Matching:
A bond is selected with a maturity that matches the last liability
Amount invested in this bond is such that the principal plus final coupon payment is equal to the last liability
The remaining elements of the liability stream are then reduced by the coupon payments of this bond
Another bond is chosen for the next to last liability and this sequence is continued until all liabilities have been matched by payments on securities selected for the portfolio

**31. **Cash Flow Matching vs. Multiple Liability Immunization Only if all liability flows were perfectly matched to the asset flows in the portfolio would there be no reinvestment risk (immunization or cash flow matching risk) . . . very unlikely
In most cases, an immunization strategy is as effective and less costly to fund liabilities. Two Factors:
1) A relatively conservative rate of return assumption for short term cash must be made throughout the life of the plan in cash flow matching
2) Funds from a cash flow matched portfolio must be available when each liability is due (usually slightly before)
Cash flow matching is a dedicated portfolio strategy

**32. **Extensions of Basic Cash Flow Matching Combination Matching or Horizon Matching
Creates a portfolio that is duration matched with the added constraint that it be cash matched in the first few years
Liquidity needs are provided for in the initial cash flow matched period
Also, since most of the positive slope or inversion of a yield curve takes place in the first few years, cash flow matching the initial portion of the liability stream reduces the risk associated with non-parallel shifts of the yield curve.
Disadvantage: Combination matching is more costly compared to a multiple liability immunization strategy